On decompositions of finite projective planes and their applications

Let PG(2; q) be the projective plane over the field Fq. Singer [19] notes that PG(2; q) has a cyclic group of order q2 + q + 1 that permutes the points of PG(2; q) in a single cycle. A k-arc set of k points no three of which are collinear. A k-arc is called complete if it is not contained in a (k + 1)-arc of PG(2; q). By taking the orbits of points under a proper subgroup of a single cycle, one can decompose the projective plane PG(2; qk) into disjoint copies of subplanes isomorphic to PG(2; q) if and only if k is not divisible by three. Moreover, by taking the orbits of points under a proper subgroup, one can decompose the projective plane PG(2; q2) into disjoint copies of complete (q2 - q + 1)-arcs. In this thesis, our main problem is to classify (up to isomorphism) the different types of decompositions of PG(2 ;q2) for q = 3; 4; 5; 7, namely subplanes and arcs. We further illustrate some of the connections between these subgeometry decompositions and other areas of combinatorial interest; in particular, we explain the relationship between coding theory and projective spaces and describe the links with Hermitian unital. Furthermore, projective codes are obtained by taking the disjoint union of such subgeometries.